# Slyn'ko V. I.

### Stability of fixed points for a class of quasilinear cascades in the space conv $R^n$

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1166-1179

The discrete dynamical systems (cascades) in semilinear metric space of nonempty convex compacts of finite-dimensional space are studied. Using the methods of convex geometry of H. Minkowski and A. D. Alexandrov the sufficient conditions of the stability of the fixed points were established. Under certain restrictions on the mappings generating the cascade, the problem of asymptotic stability of fixed point of the cascade was reduced to localization of the roots of a polynomial inside the unit circle in the complex plane. Examples of cascades in the plane were given.

### Estimates of the area of solutions of the pseudolinear differential equations with Hukuhara derivative in the space $\text{conv} (R^2)$

Ocheretnyuk E. V., Slyn'ko V. I.

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 189-214

We obtain estimates for the areas of the solutions of differential equations with Hukuhara derivative of a special form in the space $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v} (R^2)$. The main methods used for the investigation are the method of comparison, the methods of the Minkowski – Aleksandrov geometry of convex bodies, and the Chaplygin –Wa˙zewski method of approximate integration of differential equations. The obtained results enable us to reduce the estimates of the area of solutions to the investigation of differential equations of the first order.

### On the stability of abstract monotone impulsive differential equations in terms of two measures

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 904-923

We consider differential equations in a Banach space subjected to pulse influence at fixed times. It is assumed that a partial order is introduced in the Banach space with the use of a certain normal cone and that the differential equations are monotone with respect to initial data. We propose a new approach to the construction of comparison systems in a finite-dimensional space that does not involve auxiliary Lyapunov type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures, choosing a certain Birkhoff measure as the measure of initial displacements, and the norm in the given Banach space as the measure of current displacements. We give some examples of investigation of impulsive systems of differential equations in critical cases and linear impulsive systems of partial differential equations.

### Conditions for the stability of an impulsive linear equation with pure delay

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1200-1207

We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay.

### On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems

Denisenko V. S., Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 641-649

We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems.

### On stability of linear hybrid mechanical systems with distributed components

Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 204–216

We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional.